Special function (airy function) with arbitrary precision

I am solving equations which requires to evaluate Airy function and its derivative at high precision.
Although the package SpecialFunctions provides airyai and airyaiprime function,
the airyaiprime does not accept BigFloat as parameter.

I have also tried ArbNumerics. But when I do

>>>using ArbNumerics

>>>airyai(ArbFloat(0.0))

the program simply collapse and quit.

Is there any other packages which provides Airy functions (or more generally, special functions) with arbitrary precision?

It seems to take BigFloat alright in 1.10

 | | |_| | | | (_| |  |  Version 1.10.0 (2023-12-25)
 _/ |\__'_|_|_|\__'_|  |  Official https://julialang.org/ release
|__/                   |

julia> using SpecialFunctions

julia> airyai(BigFloat(0.5))
0.2316936064808334897691252545099217396183864753577499817922638265232339844931842
(@v1.10) pkg> status SpecialFunctions
[276daf66] SpecialFunctions v2.3.1
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Thank you for your reply.

sorry I made a mistake.
The airyai does accept BigFloat.
However, I also need to calculate its derivative, using airyaiprime.
It is airyaiprime that cannot calculate BigFloat.

I have already updated to julia v1.10 and SpecialFunction v2.3.1,
but it still doesn’t work.

I have modified the main post.

Works with ArbNumerics (perhaps the setprecision is needed):

julia> using ArbNumerics

julia> setprecision(ArbFloat, precision(BigFloat))
256

julia> a = ArbFloat(0.0)
0

julia> result = airyaiprime(a)
-0.25881940379280679840518356018920396347909113835493458221000181385610277267679
(@v1.10) pkg> status ArbNumerics
Status `~/.julia/environments/v1.10/Project.toml`
  [7e558dbc] ArbNumerics v1.3.3
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Thank you.

After more test, I found that ArbNumerics works fine on my Mac.
However, on my win11 laptop, airyaiprime(ArbFloat(0.0)) will lead to crash.

I think this is a bug of the package.

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Yeah, probably is a bug. Can you show the printout of the crash? Opening an issue is definitely needed.

My guess is that this is likely related to differences in Int type lengths in C (specifically, windows makes long and thus Julia’s CInt 32 bits, while on mac and linux, long is 64 bits).

Also it looks like this issue is at least reported:

The problem is, there is no printout at all.

The julia REPL simply quit without any warning.

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If you can’t get around the crash you could also consider using Arblib.jl instead. In that case you could also compute the value and derivative (and higher order derivatives) simultaneously using ArbSeries.

julia> using Arblib, SpecialFunctions

julia> airyai(Arb(0))
[0.35502805388781723926006318600418317639797917419917724058332651030081004245013 +/- 7.76e-78]

julia> airyaiprime(Arb(0))
[-0.25881940379280679840518356018920396347909113835493458221000181385610277267679 +/- 7.31e-78]

julia> airyai(ArbSeries((0, 1)))
[0.35502805388781723926006318600418317639797917419917724058332651030081004245013 +/- 7.76e-78] + [-0.25881940379280679840518356018920396347909113835493458221000181385610277267679 +/- 7.31e-78]⋅x + 𝒪(x^2)

If you don’t need the ball enclosure provided by the Arb type you can convert it to BigFloat after computing the Airy-function of course!

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